Conley conjecture for negative monotone symplectic manifolds (Q2883872)

A symplectic manifold \((M, \omega )\) is called \textit{negative monotone} if \([\omega ]=\lambda c_1(TM)\) for some \(\lambda <0\). By using the Hamiltonian Floer theory the present paper proves that a periodic Hamiltonian diffeomorphism of a closed, negative monotone symplectic manifold has infinitely many periodic orbits. In the symplectically aspherical case, the growth of periodic orbits is related to the decay of mean indices for a certain type of orbits called \textit{carriers of the action selector}. In particular, it is proved that whenever the mean indices of the carriers are bounded away from zero, the number of simple periodic orbits grows linearly with the order of iteration.